A mixed integer model for the sparsest cut problem

In a capacitated graph with a set of commodities, the sparsity of a cut is the ratio between the capacity of the cut and the demand of the commodities separated by the cut. The Sparsest Cut (SC) is often introduced as a weak dual of the Maximum Concurrent Flow problem (MCF). Contrarily to MCF, problem SC is, in general, NP-hard. This problem has been considerably studied, motivating the design of very elaborated approximation algorithms [Auman, Y. and Y. Rabani, Approximate min-cut max-flow theorem and approximations algorithm, SIAM Journal on Computing 27 (1998), pp. 213-238] [Günlük, O., On the min-cut max-flow ratio for multicommodity flows, SIAMJ. of Disc. Math. 21 (2007), pp. 1-15] [N. Garg, M. Y., V. Vazirani, Approximate max-flow min-(multi)cut theorems and their applications, in: STOC'93, 1993, pp. 698-707] [S. Plotkin, E. T., Improved bounds on the max-flow min-cut ratio for multicommodity flows, in: 25th Symposium on Theory of Computing, 1993] [Tragoudas, S., Improved approximations for the min-cut max-flow ratio and the flux, Mathematical Systems Theory 29 (1996), pp. 157-167]. Somewhat surprisingly, to the best of our knowledge, problem (SC) has not been investigated with exact approaches using Mixed Integer Programming models. In this paper, we propose a formulation arising "naturally" from the dual of (MCF).